Math 309



Math 309 -

These are notes that I have cut from another document using the TI-86 with some of the current topics.

VI. Distributions

To get the graph of the standard normal curve and shade areas to find probabilities:

i) GRAPH (r3, c1), then y(x)= (F1)

ii) Type next to y1= e^-(x^2/2)/((2()

(If you have a function in y1 that you want to save, use another storage space.)

Select [selct (F5)] shades or unshades the equal sign associated with each function. If the equal sign is shaded that function will be sketched, if not the function will be stored for later.

iii) Now we will pick the viewing screen. Press WIND for window (2nd, F2). A good window for our curve is xMin= -3

xMax= 3

xScl =1

yMin=-.1

yMax=.5

yScl=.1

This may not be a good window for other sketches!

iv) Now press GRAPH

(The(indicates that there are more options on this menu, it does not go with GRAPH.)

.

You should see the bell-shaped curve appear. After you see the sketch, press MORE (r1, c3).

iv-b. MATH (F1), then (f(x) (F3), respond to the request for a lower limit by either keying in a number or tracing using the arrow keys. When you have the desired value, press ENTER. For our example, accept 0 for the lower limit. Repeat the process for the upper limit. Let’s use 1.15 for our first example. The calculator will shade the area under the curve between your lower and upper limits. It will also give you the area that corresponds to the probability that Z falls between those values. Note that in our example, the number corresponds to the chart.

To clear the screen and find other areas, push EXIT (r1, c2), then DRAW (F2), then MORE (r1, c3) twice, then CLDRW (F1). Now you are back to iv-b.

If you want to find the tail, ANS is storing the last probability. In the last example, we found P(0 < Z < 1.15). To find P(Z > 1.15) immediately afterward, press EXIT, then .5-ANS. Compare answers with normal charts.

Obtaining values of normal distribution tables without sketching the graph:

i) Press CALC [ 2nd (], then press fnInt [F5]. After the “fnInt(“ prompt , list the function, the independent variable, the lower limit, the upper limit. In our example, fnInt(y1,x,0,1.15). Press ENTER.

ii) Use ENTRY [ 2nd ENTER ] to repeat the line. Edit the upper and lower limits with the arrow keys.

To find values for non-standard normal distributions:

Enter in the density function in the “y(x)=” list. For example, y2= e^-((x-( )^2/(2(2))/(( ((2()) where you put in the mean and standard deviation of your distribution for ( and (. Select this function, and deselect any others. Now repeat the above steps.

Other continuous distributions can be done similarly by using their density functions in place of

e^-(x^2/2)/((2()

APPLICATION: Appendix F, Statistics for the Terrified.

Discrete distributions (tables)

Values from tables for such distributions as the binomial and Poisson can easily be computed on the calculator using their distributions.

For example, let X be a binomial random variable with n = 10, p = 0.7.

You can get the probability distribution for X by:

seq(10 nCr x * .7^x*.3^(10-x),x,0,10)

The keystrokes to get this command are:

seq MATH (2nd, r6, c5), MISC (F5), F3

( (r5, c3); 10

nCr MATH (2nd, r6, c5), PROB (F2), F3

x (r2, c2)

then use the lower right corner calculator keys to finish the line

Note that: P(X=6) =10 nCr 6 * .7^6*.3^4

P(2 (X(5) = sum seq(10 nCr x * .7^x*.3^(10-x),x,2,5)

where “sum” is MATH (2nd, r6, c5), MISC (F5), F1.

Check the computation by sum seq(10 nCr x * .7^x*.3^(10-x),x,0,10)

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